1 FINITE POINT CONFIGURATIONS J´Anos Pach

1 FINITE POINT CONFIGURATIONS János Pach INTRODUCTION The study of combinatorial properties of finite point configurations is a vast area of research in geometry, whose origins go back at least to the ancient Greeks. Since it includes virtually all problems starting with “consider a set of n points in space,” space limitations impose the necessity of making choices. As a result, we will restrict our attention to Euclidean spaces and will discuss problems that we find particularly important. The chapter is partitioned into incidence problems (Section 1.1), metric problems (Section 1.2), and coloring problems (Section 1.3). 1.1 INCIDENCE PROBLEMS In this section we will be concerned mainly with the structure of incidences between a finite point configuration P and a set of finitely many lines (or, more generally, k- dimensional flats, spheres, etc.). Sometimes this set consists of all lines connecting the elements of P . The prototype of such a question was raised by Sylvester [Syl93] more than one hundred years ago: Is it true that for any configuration of finitely many points in the plane, not all on a line, there is a line passing through exactly two points? This problem was rediscovered by Erd˝os [Erd43], and affirmative answers to this question was were given by Gallai and others [Ste44]. Generalizations for circles and conic sections in place of lines were established by Motzkin [Mot51] and Wilson-Wiseman [WW88], respectively. GLOSSARY Incidence: A point of configuration P lies on an element of a given collection of lines (k-flats, spheres, etc.). Simple crossing: A point incident with exactly two elements of a given collection of lines or circles. Ordinary line: A line passing through exactly two elements of a given point configuration. Ordinary circle: A circle passing through exactly three elements of a given point configuration. Ordinary hyperplane: A (d 1)-dimensional flat passing through exactly d elements of a point configuration− in Euclidean d-space. Motzkin hyperplane: A hyperplane whose intersection with a given d-dimensional point configuration lies—with the exception of exactly one point—in a (d 2)-dimensional flat. − 3 Preliminary version (August 10, 2017). To appear in the Handbook of Discrete and Computational Geometry, J.E. Goodman, J. O'Rourke, and C. D. Tóth (editors), 3rd edition, CRC Press, Boca Raton, FL, 2017. 4 J. Pach Family of pseudolines: A family of two-way unbounded Jordan curves, any two of which have exactly one point in common, which is a proper crossing. Family of pseudocircles: A family of closed Jordan curves, any two of which have at most two points in common, at which the two curves properly cross each other. Regular family of curves: A family Γ of curves in the xy-plane defined in terms of D real parameters satisfying the following properties. There is an integer s such that (a) the dependence of the curves on x, y, and the parameters is algebraic of degree at most s; (b) no two distinct curves of Γ intersect in more than s points; (c) for any D points of the plane, there are at most s curves in Γ passing through all of them. Degrees of freedom: The smallest number D of real parameters defining a regular family of curves. Spanning tree: A tree whose vertex set is a given set of points and whose edges are line segments. Spanning path: A spanning tree that is a polygonal path. Convex position: P forms the vertex set of a convex polygon or polytope. k-Set: A k-element subset of P that can be obtained by intersecting P with an open halfspace. Halving plane: A hyperplane with P /2 points of P on each side. ⌊| | ⌋ SYLVESTER-TYPE RESULTS 1. Gallai theorem (dual version): Any set of lines in the plane, not all of which pass through the same point, determines a simple crossing. This holds even for families of pseudolines [KR72]. 2. Pinchasi theorem: Any set of at least five pairwise crossing unit circles in the plane determines a simple crossing [Pin01]. Any sufficiently large set of pairwise crossing pseudocircles in the plane, not all of which pass through the same pair of points, determines an intersection point incident to at most three pseudocircles [ANP+04]. 3. Pach-Pinchasi theorem: Given n red and n blue points in the plane, not all on a line, there always exists a bichromatic line containing at most two points of each color [PP00]. Any finite set of red and blue points contains a monochromatic spanned line, but not always a monochromatic ordinary line [Cha70]. 4. Motzkin-Hansen theorem: For any finite set of points in Euclidean d-space, not all of which lie on a hyperplane, there exists a Motzkin hyperplane [Mot51, Han65]. We obtain as a corollary that n points in d-space, not all of which lie on a hyperplane, determine at least n distinct hyperplanes. (A hyperplane is determined by a point set P if its intersection with P is not contained in a (d 2)-flat.) Putting the points on two skew lines in 3-space shows that the existence− of an ordinary hyperplane cannot be guaranteed for d> 2. Preliminary version (August 10, 2017). To appear in the Handbook of Discrete and Computational Geometry, J.E. Goodman, J. O'Rourke, and C. D. Tóth (editors), 3rd edition, CRC Press, Boca Raton, FL, 2017. Chapter 1: Finite point configurations 5 If n > 8 is sufficiently large, then any set of n noncocircular points in the n 1 plane determines at least −2 distinct circles, and this bound is best possible [Ell67]. The minimum number of ordinary circles determined by n noncocircular points is 1 n2 O(n). Here lines are not counted as circles [LMM+17]. 4 − 5. Csima-Sawyer theorem: Any set of n noncollinear points in the plane determines at least 6n/13 ordinary lines (n > 7). This bound is sharp for n = 13 and false for n = 7 [KM58, CS93]; see Figure 1.1.1. Green-Tao theorem (formerly Motzkin-Dirac conjecture): If n is sufficiently large and even, then the number of ordinary lines is at least n/2, and if n n 1 is odd, then at least 3 −4 . These bounds cannot be improved [GT13]. In 3-space, any set of n noncoplanar⌊ ⌋ points determines at least 2n/5 Motzkin hyperplanes [Han80, GS84]. FIGURE 1.1.1 Extremal examples for the (dual) Csima-Sawyer theorem: (a) 13 lines (including the line at infinity) determining only 6 simple points; (b) 7 lines determining only 3 simple points. (a) (b) 6. Orchard problem [Syl67]: What is the maximum number of collinear triples determined by n points in the plane, no four on a line? There are several constructions showing that this number is at least n2/6 O(n), which is asymptotically best possible, cf. [BGS74, FP84]. (See Figure− 1.1.2.) Green- Tao theorem [GT13]: If n is sufficiently large, then the precise value of the n(n 3) maximum is − . ⌊ 6 ⌋ FIGURE 1.1.2 L 12 points and 19 lines, each passing through exactly 3 points. 7. Dirac’s problem [Dir51]: Does there exist a constant c0 such that any set of n points in the plane, not all on a line, has an element incident to at least n/2 c connecting lines? If true, this result is best possible, as is − 0 Preliminary version (August 10, 2017). To appear in the Handbook of Discrete and Computational Geometry, J.E. Goodman, J. O'Rourke, and C. D. Tóth (editors), 3rd edition, CRC Press, Boca Raton, FL, 2017. 6 J. Pach shown by the example of n points distributed as evenly as possible on two intersecting lines. (It was believed that, apart from some small examples listed in [Grü72], this statement is true with c0 = 0, until Felsner exhibited an infinite series of configurations, showing that c0 3/2; see [BMP05, p. 313].) It is known [PW14] that there is a positive constant≥ c such that one can find a point incident to at least cn connecting lines. The best known value of the constant for which this holds is c =1/37. A useful equivalent formulation of this assertion is that any set of n points in the plane, no more than n k of − which are on the same line, determines at least c′kn distinct connecting lines, for a suitable constant c′ > 0. Note that according to the d = 2 special case of the Motzkin-Hansen theorem, due to Erd˝os (see No. 4 above), for k = 1 the number of distinct connecting lines is at least n. For k = 2, the corresponding bound is 2n 4, (n 10). − ≥ 8. Ungar’s theorem [Ung82]: n noncollinear points in the plane always determine at least 2 n/2 lines of different slopes (see Figure 1.1.3); this proves Scott’s conjecture.⌊ Furthermore,⌋ any set of n points in the plane, not all on a line, permits a spanning tree, all of whose n 1 edges have different slopes [Jam87]. Pach, Pinchasi, and Sharir [PPS07] proved− Scott’s conjecture in 3 dimensions: any set of n 6 points in R3, not all of which are on a plane, determine at least 2n 5 pairwise≥ nonparallel lines if n is odd, and at least 2n 7 if n is even. This− bound is tight for every odd n. − FIGURE 1.1.3 7 points determining 6 distinct slopes. UPPER BOUNDS ON THE NUMBER OF INCIDENCES Given a set P of n points and a family Γ of m curves or surfaces, the number of incidences between them can be obtained by summing over all p P the number of elements of Γ passing through p.

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